Bulletin T.CXLIV de l’Académie serbe des sciences et des arts − 2012
Classe des Sciences mathématiques et naturelles
Sciences mathématiques, No 37
CONFORMAL CHANGE OF THE METRIC ON ALMOST ANTI-HERMITIAN
MANIFOLDS
MILEVA PRVANOVIĆ
(Presented at the 1st Meeting, held on February 24, 2012)
A b s t r a c t. Some conformally invariant algebraic curvature tensors
for the almost anti-Hermitian manifolds are found. It is proved that they
are related such that the equation (7.4) holds.
AMS Mathematics Subject Classification (2000): 53C56
Key Words: anti-Hermitian manifold, anti-Kähler manifold, conformally
related metrics, conformally invariant tensors
1. Intoduction
An almost anti-Hermitian manifold (M, g, J) is a differentiable manifold
M, dim M = 2n, endowed with complex structure J and anti-Hermitian
metric g, i.e.
J 2 = −Id., g(JX, JY ) = −g(X, Y )
(1.1)
for any vector fields X, Y of the tangent vector space T (M ). Then
F (X, Y ) = g(JX, Y ) = g(X, JY ) = F (X, Y )
and
F (JX, JY ) = −F (X, Y ).
40
Mileva Prvanović
If {ei } i = 1, 2, . . . , 2n is an orthonormal basis of Tp (M ), p ∈ M, and
R(X, Y, Z, W ) is the Riemannian curvature tensor, then the first and the
second Ricci tensors are respectively
ρ(Y, Z) =
2n
∑
R(ei , Y, Z, ei ),
ρe(Y, Z) =
i=1
2n
∑
R(Jei , Y, Z, ei ),
i=1
while the first and the second scalar curvatures are
κ=
2n
∑
ρ(ei , ei ),
e=
κ
i=1
2n
∑
ρe(ei , ei ).
i=1
We note that, like the first, the second Ricci tensor is symmetric, that
is ρe(Y, Z) = ρe(Z, Y ).
Let ∇ is the Levi-Civita connection with respect to the metric g. If
besides (1.1), the condition ∇J = 0 is also satisfied, we have
R(X, Y, JZ, JW ) = −R(X, Y, Z, W ).
(1.2)
It is well known that the condition R(X, Y, JZ, JW ) = R(X, Y, Z, W )
is characteristic for the Kähler manifolds. Thus, the anti-Hermitian manifold satisffying the condition ∇J = 0, is called anti-Kähler manifold or
the Kähler manifold with Norden metric. Such manifolds were first investigated by A.P.Norden [7] in the case dim M = 4. He named them B−manifolds
to distinguish them from the Kähler spaces (A−manifolds in [7]).
The relation (1.2) implies [8]:
R(JX, JY, JZ, JW ) = R(X, Y, Z, W )
(1.3)
and
R(JX, Y, Z, W ) = R(X, JY, Z, W ) = R(X, Y, JZ, W ) = R(X, Y, Z, JW ).
(1.4)
The Ricci tensors of the anti-K̈ahler manifold satisfy the conditions
ρ(JX, Y ) = ρe(X, Y ),
ρ(X, Y ) = −ρe(JX, Y ),
ρ(JX, JY ) = −ρ(X, Y ), ρe(JX, JY ) = −ρe(X, Y ).
Indeed, in view of (1.4), we have
J(JX, Y ) =
2n
∑
i=1
R(ei , JX, Y, ei ) =
2n
∑
i=1
R(Jei , X, Y, ei ) = ρe(X, Y ),
(1.5)
41
Conformal change of the metric on almost anti-Hermitian manifolds
ρe(JX, Y ) =
2n
∑
R(Jei , JX, Y, ei ) =
i=1
2n
∑
=−
2n
∑
R(J 2 ei , X, Y, ei )
i=1
R(ei , X, Y, ei ) = −ρ(X, Y ),
i=1
and similarly for other relations (1.5). Also,
∑
i
∑
ρ(Jei , ei ) =
∑
e,
ρe(ei , ei ) = κ
i∑
ρe(Jei , ei ) = −
i
ρ(ei , ei ) = −κ.
(1.6)
i
The anti-Kähler and anti-Hermitian manifolds have been investigated
relatively intensively for last ten to fiftien years ([1],[2],[4],[5],[8],[9],[10]).
The object of the present paper is to investigate the effects of the conformal
change of the metric. This will be done by using the tensor defined in the
section 2. The first results are exhibited in the section 3. In the sections
4, 5 and 6 we obtain conformally invariant algebraic curvature tensors, and
in the section 7 we prove that they are related such that the equation (7.4)
holds.
2. Algebraic curvature tensor satisfying the condition of type (1.2)
If ∇J ̸= 0, the anti-Kähler condition (1.2) does not hold. Yet, for any
almost anti-Hermitian manifold there exists the algebraic curvature tensor
satisfying the condition of the type (1.2). It is
H(X, Y, Z, W ) =
1
8
[R(X, Y, Z, W ) + R(JX, JY, JZ, JW )
−R(JX, JY, Z, W ) − R(JX, Y, JZ, W ) − R(JX, Y, Z, JW )
(2.1)
−R(X, JY, JZ, W ) − R(X, JY, Z, JW ) − R(X, Y, JZ, JW )] .
By the direct calculation, we can see that
H(X, Y, Z, W ) = −H(Y, X, Z, W ) = −H(X, Y, W, Z) = H(Z, W, X, Y ),
H(X, Y, Z, W ) + H(Y, Z, X, W ) + H(Z, X, Y, W ) = 0,
(2.2)
as well as
H(X, Y, JZ, JW ) = −H(X, Y, Z, W )
H(JX, Y, Z, W ) = H(X, JY, Z, W ) = H(X, Y, JZ, W ) = H(X, Y, Z, JW ).
(2.3)
42
Mileva Prvanović
The relations (2.2) show that the tensor H(X, Y, Z, W ) is the algebraic
curvature tensor (that is, has all algebraic properties as the Riemannian curvature tensor R(X, Y, Z, W )) while (2.3) show that is satisfies the condition
of type (1.2).
We note that for anti-Kähler manifolds, i.e. if ∇J = 0, it holds
H(X, Y, Z, W ) = R(X, Y, Z, W ).
(2.4)
The first Ricci tensor, corresponding to the tensor H(X, Y, Z, W ) is
ρ(H)(Y, Z) =
∑
H(ei , Y, Z, ei )
i
=
1
{ρ(Y, Z) − ρ(JY, JZ)
8
−ρe(JY, Z) − ρe(Y, JZ) + ρ(Y, Z)
−ρ(JY, JZ) − ρe(Z, JY ) − ρe(JZ, Y )} .
Thus, and in view of the symmetry of the tensor ρe(Y, Z), we have
ρ(H)(Y, Z) =
1
{ρ(Y, Z) − ρ(JY, JZ) − ρe(Y, JZ) − ρe(JY, Z)} .
4
(2.5)
In the similar way we obtain
1
ρe(H)(Y, Z) = {ρe(Y, Z) − ρe(JY, JZ) + ρ(Y, JZ) + ρ(JY, Z)}.
4
(2.6)
Therefore
ρ(H)(JY, Z) = ρe(H)(Y, Z),
ρe(H)(JY, Z) = −ρ(H)(Y, Z),
ρ(H)(JY, JZ) = −ρ(H)(Y, Z), ρe(H)(JY, JZ) = −ρe(H)(Y, Z),
(2.7)
that is, ρ(H) and ρe(H) satisfy the conditions analogous to the conditions
(1.5).
Finally,
]
[
∑
1
ρe(ei , Jei ) ,
κ(H) =
κ−
2
i
[
]
(2.8)
∑
1
e (H) =
e+
κ
κ
ρ(ei , Jei ) .
2
i
43
Conformal change of the metric on almost anti-Hermitian manifolds
3. Conformal change of the metric
Now, let us consider the conformal change of the metric
ḡ = e2f g,
¯ the Levi-Civita connection with
where f is a scalar function. Denoting by ∇
respect to the metric ḡ, we have
¯ − ∇)(X, Y ) = θ(X)(Y ) + θ(Y )(X) − g(X, Y )U,
(∇
for any vector fields X, Y ∈ T (M ), where θ is l-form defined by θ = df, and
U is the vector field such that g(U, X) = θ(X).
From now on, all geometric objects in (M, ḡ, J) will be denoted by analogous letters as in (M, g, J), but with ”bar”.
It is well known that the Riemanian curvature tensors R̄ and R of the
metrics ḡ and g respectively, are related as follows (see for ex. [6]).
e−2f R̄(X, Y, Z, W ) = R(X, Y, Z, W )+
+g(X, W )σ(Y, Z) + g(Y, Z)σ(X, W ) − g(X, Z)σ(Y, W ) − g(Y, W )σ(X, Z),
(3.1)
where σ is the tensor field of type (0,2) defined by
1
σ(X, Y ) = (∇X θ)(Y ) − θ(X)θ(Y ) + θ(U )g(X, Y ).
2
We note that σ(X, Y ) = σ(Y, X) because θ is a gradient.
It follows from (3.1)
ρ̄(Y, Z) = ρ(Y, Z) + 2(n − 1)σ(Y, Z) + g(Y, Z)
∑
σ(ei , ei ).
i
Therefore
∑
σ(ei , ei ) =
i
e2f κ̄ − κ
,
2(2n − 1)
(3.2)
such that
[
]
1
κ̄
σ(Y, Z) =
ρ̄(Y, Z) −
ḡ(Y, Z)
2(n − 1)
2(2n − 1)
[
]
1
κ
−
ρ(Y, Z) −
g(Y, Z) .
2(n − 1)
2(2n − 1)
(3.3)
44
Mileva Prvanović
This relation, together with
[
]
1
κ̄
σ(JY, JZ) =
ρ̄(JY, JZ) +
ḡ(Y, Z)
2(n − 1)
2(2n − 1)
[
]
1
κ
−
ρ(Jy, JZ) +
g(Y, Z) ,
2(n − 1)
2(2n − 1)
yields
1
κ̄
[ρ̄(Y, Z) − ρ̄(JY, JZ)] −
ḡ(Y, Z)
2(n − 1)
2(n − 1)(2n − 1)
1
κ
−
[ρ(Y, Z) − ρ(JY, Jz)] +
g(Y, Z),
2(n − 1)
2(n − 1)(2n − 1)
S(Y, Z) =
(3.4)
where we have put
S(Y, Z) = σ(Y, Z) − σ(JY, JZ).
(3.5)
We note that S(JY, JZ) = −S(Y, Z), and thus S(Y, JZ) = S(JY, Z).
On the other hand, putting into (3.1), X = Jei , W = ei and summing
up, we obtain
ē
ρ(Y,
Z) = ρe(Y, Z) + g(Y, Z)
∑
σ(ei , Jei ) − [σ(Y, JZ) + σ(JY, Z)] ,
(3.6)
i
where from it follows
∑
σ(ei , Jei ) =
i
ē − κ
e
e2f κ
,
2(n − 1)
(3.7)
such that (3.6) becomes
ē
σ(Y, Z) − σ(JY, JZ) = ρ(JY,
Z) −
[
− ρe(JY, Z) −
ē
κ
F̄ (Y, Z)
2(n − 1)
]
e
κ
F (Y, Z) .
2(n − 1)
The symmetric part of this relation, in view of (3.5), is
]
ē
1[
κ
ē
ē
ρ(Y,
JZ) + ρ(JY,
Z) −
F̄ (Y, Z)
2
2(n − 1)
e
1
κ
− [ρe(Y, JZ) + ρe(JY, Z)] +
F (Y, Z).
2
2(n − 1)
S(Y, Z) =
(3.8)
45
Conformal change of the metric on almost anti-Hermitian manifolds
Comparating (3.4) and (3.8), we find
[
]
κ̄
ē F̄ (Y, Z)
ḡ(Y, Z) + κ
2n − 1
κ
e F (Y, Z).
= ρ(Y, Z) − ρ(JY, JZ) − (n − 1) [ρe(Y, JZ) + ρe(JY, Z)] −
g(Y, Z) + κ
2n − 1
(3.9)
Thus, we can state
ē
ē
ρ̄(Y, Z) − ρ̄(JY, JZ) − (n − 1) ρ(Y,
JZ) + ρ(JY,
Z) −
Proposition 3.1 For an almost anti-Hermitian manifold, the tensor
P (Y, Z) = ρ(Y, Z) − ρ(JY, JZ) − (n − 1) [ρe(Y, JZ) + ρe(JY, Z)]
−
(3.10)
κ
e F (Y, Z)
g(Y, Z) + κ
2n − 1
is conformally invariant.
For the later use, we present still two other forms of the tensor (3.10).
Putting into (3.9) Y = Z = ei and summing up, we obtain
[
e
2f
]
∑
∑
κ̄
κ
ē i , Jei ) =
−
−
ρ(e
ρe(ei , Jei ),
2n − 1
2n
−
1
i
i
(3.11)
while putting Y = ei , Z = Jei , we get
[
ē −
e2f κ
∑
]
e−
ρ̄(ei , Jei ) = κ
∑
i
ρ(ei , Jei ).
(3.12)
i
The relation (3.12) implies
ē F̄ (Y, Z) − F̄ (Y, Z)
κ
∑
e F (Y, Z) − F (Y, Z)
ρ̄(ei , Jei ) = κ
i
∑
ρ(ei , Jei ).
i
Substituting this into (3.9), we find
[
]
ē
ē
ρ̄(Y, Z)−ρ̄(JY, JZ)−(n−1) ρ(Y,
JZ) + ρ(JY,
Z) −
∑
κ̄
ḡ(Y, Z)+F̄ (Y, Z)
ḡ(ei , Jei )
2n − 1
i
= ρ(Y, Z)−ρ(JY, JZ)−(n−1) [ρe(Y, JZ) + ρe(JY, Z)]−
∑
κ
g(Y, Z)+F (Y, Z)
ρ(ei , Jei ).
2n − 1
i
Similarly, the relation (3.11) yields
−
∑
∑
κ
κ̄
ē i , Jei )−
ρ(e
ρe(ei , Jei ).
ḡ(Y, Z) = −ḡ(Y, Z)
g(Y, Z)+g(Y, Z)
2n − 1
2n − 1
i
i
46
Mileva Prvanović
Substituting this into (3.9), we obtain
[
]
ē
ē
ρ̄(Y, Z) − ρ̄(JY, JZ) − (n − 1) ρ(Y,
JZ) + ρ(JY,
Z)
−ḡ(Y, Z)
∑
ē i , Jei ) + κ
ē F̄ (Y, Z)
ρ(e
i
= ρ(Y, Z) − ρ(JY, JZ) − (n − 1) [ρe(Y, JZ) + ρe(JY, Z)]
−g(Y, Z)
∑
e F (Y, Z).
ρe(ei , Jei ) + κ
i
Thus, we can state
Proposition 3.2 The tensors
P ′ (Y, Z) = ρ(Y, Z) − ρ(JY, JZ) − (n − 1) [ρe(Y, JZ) + ρe(JY, Z)]
−
∑
κ
g(Y, Z) + F (Y, Z)
ρ(ei , Jei ).
2n − 1
i
(3.13)
and
P ′′ (Y, Z) = ρ(Y, Z) − ρ(JY, JZ) − (n − 1) [ρe(Y, JZ) + ρe(JY, Z)]
−g(Y, Z)
∑
(3.14)
e F (Y, Z).
ρe(ei , Jei ) + κ
i
are conformally invariant.
4. The first conformally invariant algebraical curvature tensor
The algebraic curvature tensor (2.1) with respect to the metric ḡ is
[
8e−2f H̄(X, Y, Z, W )= e−2f R̄(X, Y, Z, W ) + R̄(JX, JY, JZ, JW )
−R̄(JX, JY, Z, W ) − R̄(JX, Y, JZ, W ) − R̄(JX, Y, Z, JW )
]
−R̄(X, JY, JZ, W ) − R̄(X, JY, Z, JW ) − R̄(X, Y, JZ, JW )
Substituting (3.1) and using the notation (3.5), we get
e−2f H̄(X, Y, Z, W ) = H(X, Y, Z, W )
1
+ [g(X, W )S(Y, Z) + g(Y, Z)S(X, W ) − g(X, Z)S(Y, W )
4
−g(Y, W )S(X, Z) − F (X, W )S(Y, JZ) − F (Y, Z)S(X, JW )
+F (X, Z)S(Y, JW ) + F (Y, W )S(X, JZ)]
(4.1)
Conformal change of the metric on almost anti-Hermitian manifolds
47
Putting X = W = ei , we find
4ρ(H̄)(Y, Z) = 4ρ(H)(Y, Z) + 2(n − 2)S(Y, Z)
+g(Y, Z)
∑
S(ei , ei ) − F (Y, Z)
i
∑
S(ei , Jei ),
(4.2)
i
where from, putting Y = Z = ei , we obtain
∑
S(ei , ei ) =
i
To find
∑
e2f κ(H̄) κ(H)
−
.
n−1
n−1
S(ei , Jei ), we put into (4.2) Y = Jei , Z = ei , and get
i
∑
S(ei , Jei ) =
i
e (H̄)
e (H)
e2f κ
κ
−
.
n−1
n−1
Thus, the relation (4.2), for n > 2, yields
[
]
e (H̄)
1
1
1
κ(H̄)
κ
S(Y, Z) =
ρ(H̄)(Y, Z) −
ḡ(Y, Z) +
F̄ (Y, Z)
4
n−2 2
8(n − 1)
8(n − 1)
]
[
e (H)
1
1
κ(H)
κ
ρ(H)(Y, Z) −
g(Y, Z) +
F (Y, Z) .
−
n−2 2
8(n − 1)
8(n − 1)
(4.3)
Finally, substituting (4.3) into (4.1), we have, for n > 2,
e−2f B̄(X, Y, Z, W ) = B(X, Y, Z, W ),
1
1
48
Mileva Prvanović
where
B(X, Y, Z, W ) = H(X, Y, Z, W )
1
1
−
[g(X, W )ρ(H)(Y, Z) + g(Y, Z)ρ(H)(X, W )
2(n − 2)
−g(X, Z)ρ(H)(Y, W ) − g(Y, W )ρ(H)(X, Z)
−F (X, W )ρ(H)(Y, JZ) − F (Y, Z)ρ(H)(X, JW )
+F (X, Z)ρ(H)(Y, JW ) + F (Y, W )ρ(H)(X, JZ)]
(4.4)
κ(H)
+
[g(X, W )g(Y, Z) − g(X, Z)g(Y, W )
4(n − 1)(n − 2)
−F (X, W )F (Y, Z) + F (X, Z)F (Y, W )]
−
e (H)
κ
[g(X, W )F (Y, Z) + g(Y, Z)F (X, W )
4(n − 1)(n − 2)
−g(X, Z)F (Y, W ) − g(Y, W )F (X, Z)] ,
and B̄(X, Y, Z, W ) is constructed in the same way, but with respect to the
1
metric ḡ.
The tensor (4.4) is the algebraic curvature tensor. Also, it satisfies the
condition of type (1.2). We say that the tensor (4.4) is the first conformally
invariant algebraic curvature tensor of the almost anti-Hermitian manifold
(M, g, J).
For the first Ricci tensor corresponding to the tensor (4.4), we have
ρ( B)(Y, Z) =
1
1
=
2(n − 2)
∑
[ i
∑
B(ei , Y, Z, ei )
1
]
e (H) F (Y, Z).
ρ(H)(ei , Jei ) − κ
i
But, according (2.7),
∑
ρ(H)(ei , Jei ) =
i
∑
e (H).
ρe(H)(ei , ei ) = κ
i
Therefore, ρ( B)(Y, Z) = 0. In the similar way we prove that ρe( B)(Y, Z) =
1
0. Thus, we can state
1
Theorem 4.1. For an almost anti-Hermitian manifold (M, g, J), dim M >
Conformal change of the metric on almost anti-Hermitian manifolds
49
4, the tensor (4.4) is the first conformally invariant algebraic curvature tensor. Both its Ricci tensors vanish.
The Ricci tensor of the generalized Bochner curvature tensor of an almost Hermitian manifold vanishes. Thus, we can say that for an almost
anti-Hermitian manifold, the tensor (4.4) is the tensor corresponding to the
generalized Bochner curvature tensor.
For an anti-Kähler manifold, the relation (2.4) holds, such that the tensor
B has the form
1
B(X, Y, Z, W ) = R(X, Y, Z, W )
1
[g(X, W )ρ(Y, Z) + g(Y, Z)ρ(X, W ) − g(X, Z)ρ(Y, W )
2(n − 2)
−g(Y, W )g(X, Z) − F (X, W )ρ(Y, JZ) − F (Y, Z)ρ(X, JW )
−
+F (X, Z)ρ(Y, JW ) + F (Y, W )ρ(X, JZ)]
κ
[g(X, W )g(Y, Z) − g(X, Z)g(Y, W )
4(n − 1)(n − 2)
−F (X, W )F (Y, Z) + F (X, Z)F (Y, W )]
(4.5)
+
e
κ
[g(X, W )F (Y, Z) + g(Y, Z)F (X, W )
4(n − 1)(n − 2)
−g(X, Z)F (Y, W ) − g(Y, W )F (X, Z)] .
−
But this is just the tensor obtained in [8] using the pseudoconformal
correspondence ḡ = αg + βF, where α and β are some scalar function.
5. The second conformally invariant algebraic curvature tensor
Substituting (3.4) into (4.1) and putting
50
Mileva Prvanović
B(X, Y, Z, W ) = H(X, Y, Z, W )
2
1
−
{g(X, W ) [ρ(Y, Z) − ρ(JY, JZ)] + g(Y, Z) [ρ(X, W ) − ρ(JX, JW )]
8(n − 1)
−g(X, Z) [ρ(Y, W ) − ρ(JY, JW )] − g(Y, W ) [ρ(X, Z) − ρ(JX, JZ)]
−F (X, W ) [ρ(Y, JZ) + ρ(JY, Z)] − F (Y, Z) [ρ(X, JW ) + ρ(JX, W )]
+F (X, Z) [ρ(Y, JW ) + ρ(JY, W )] + F (Y, W ) [ρ(X, JZ) + ρ(JX, Z)]}
+
κ
[g(X, W )g(Y, Z) − g(X, Z)g(Y, W )
4(n − 1)(n − 2)
−F (X, W )F (Y, Z) + F (X, Z)F (Y, W )] ,
(5.1)
we see at once that
e−2f B̄(X, Y, Z, W ) = B(X, Y, Z, W ).
2
2
The tensor (5.1) is the algebraic curvature tensor and it satisfies the
condition of type (1.2). We say that (5.1) is the second conformall invariant
algebraic curvature tensor of the almost anti-Hermitian manifold.
The tensor (5.1) can also be obtained in the following way.
It is well known that the Weyl tensor of conformal curvature for a Riemannian manifold (M, g), dim M = 2n, is (see for ex. [6])
C(X, Y, Z, W ) = R(X, Y, Z, W )
1
[g(X, W )ρ(Y, Z) + g(Y, Z)ρ(X, W ) − g(X, Z)ρ(Y, W ) − g(Y, W )ρ(X, Z)]
2(n − 1)
κ
+
[g(Y, W )g(Y, Z) − g(X, Z)f (Y, W )] .
2(n − 1)(2n − 1)
−
This can be rewritten in the form
1
κ
C =R−
φ+
π
2(n − 1)
2(n − 1)(2n − 1)
(5.2)
where
φ(X, Y, Z, W ) = g(X, W )ρ(Y, Z) + g(Y, Z)ρ(X, W )
−g(X, Z)ρ(Y, W ) − g(Y, W )ρ(X, Z),
π(X, Y, Z, W ) = g(X, W )g(Y, Z) − g(X, Z)g(Y, W ).
(5.3)
51
Conformal change of the metric on almost anti-Hermitian manifolds
In [11], G. Stanilov used the holomorphic curvature tensor of an almost
Hermitian manifold and applied it as an operator to the tensor (5.2). Here,
we use the tensor (2.1). Applying it to the tensor (5.2) instead to the tensor
R, we get
H(C) = H(R) −
1
κ
H(φ) +
H(π).
16(n − 1)
16(n − 1)(2n − 1)
(5.4)
In view of (5.3) we find
H(φ)(X, Y, Z, W ) =
2 {g(X, W )[ρ(Y, Z) − ρ(JY, JZ)] + g(Y, Z)[ρ(X, W ) − ρ(JX, JW )]
−g(X, Z)[ρ(Y, W ) − ρ(JY, JW )] − g(Y, W )[ρ(X, Z) − ρ(JX, JZ)]
−F (X, W )[ρ(Y, JZ) + ρ(JY, Z)] − F (Y, Z)[ρ(X, JW ) + ρ(JX, W )]
+F (X, Z)[ρ(Y, JW ) + ρ(JY, W )] + F (Y, W )[ρ(X, JZ) + ρ(JX, Z)]}
H(π)(X, Y, Z, W ) = 2[g(X, W )g(Y, Z) − g(X, Z)g(Y, W )
−F (X, W )F (Y, Z) + F (X, Z)F (Y, W )].
Thus, the right hand side of the relation (5.4) is just the right hand side
of the relation (5.1), that is, we have
H(C)(X, Y, Z, W ) = B(X, Y, Z, W ).
2
(5.5)
The first Ricci tensor corresponding to the tensor (5.1) is
ρ( B)(Y, Z) =
2
∑
i
B(ei , Y, Z, ei )
2
n−2
= ρ(H)(Y, Z) −
[ρ(Y, Z) − ρ(JY, JZ)]
4(n − 1)
∑
κ
1
−
g(Y, Z) +
F (Y, Z)
ρ(ei , Jei )
4(n − 1)(2n − 1)
4(n − 1)
i
or, in view of (2.5),
ρ( B)(Y, Z) =
2
1
{ρ(Y, Z) − ρ(JY, JZ) − (n − 1) [ρe(Y, JZ) + ρe(JY, JZ)]
4(n − 1)
}
∑
κ
−
g(Y, Z) + F (Y, Z)
ρ(ei , Jei )
2n − 1
i
(5.6)
52
Mileva Prvanović
Comparing the relations (5.6) and (3.13), we see that
ρ( B)(Y, Z) =
2
1
P ′ (Y, Z).
4(n − 1)
(5.7)
Setting into (5.1) X = Jei , W = ei , and using (2.6), we get
ρe( B)(Y, Z) =
2
1
{ρ(Y, JZ) + ρ(JY, Z) + (n − 1) [ρe(Y, Z) − ρe(JY, JZ)]
4(n − 1)
−g(Y, Z)
∑
i
}
κ
F (Y, Z) .
ρ(ei , Jei ) −
2n − 1
(5.8)
The relations (5.6) and (5.8) show that
ρ( B)(Y, JZ) = ρe( B)(Y, Z).
2
(5.9)
2
Thus, we can state
Theorem 5.1 For almost anti-Hermitian manifold, the tensor (5.1) is
the second conformally invariant algebraic curvature tensor. It satisfies the
condition (5.5). The relations (5.6) and (5.8) determine its Ricci tensors
and (5.7) and (5.9) hold.
We remark that the relation (5.9) is also the consequence of the fact that
the tensor (5.1) satisfies the condition of the type (1.2).
It (M, g, J) is anti-Kähler manifold, then (1.5) and (2.4) hold, such that
B(X, Y, Z, W ) = R(X, Y, Z, W )
2
1
[g(X, W )ρ(Y, Z) + g(Y, Z)ρ(X, W )
−
4(n − 1)
−g(X, Z)ρ(Y, W ) − g(Y, W )ρ(X, Z)
−F (X, W )ρ(Y, JZ) − F (Y, Z)ρ(X, JW )
(5.10)
+F (X, Z)ρ(Y, JW ) + F (Y, W )ρ(X, JZ)]
+
κ
[g(X, W )g(Y, Z) − g(X, Z)g(Y, W )
4(n − 1)(2n − 1)
−F (X, W )F (Y, Z) + F (X, Z)F (Y, W )],
and
[
ρ( B) =
2
]
n
1
κ
e F (Y, Z)
ρ(Y, Z) −
g(Y, Z) − κ
2(n − 1)
4(n − 1) 2n − 1
(5.11)
53
Conformal change of the metric on almost anti-Hermitian manifolds
6. The third conformally invariant algebraic curvature tensor
Substituting (3.8) into (4.1), we obtain
e−2f B̄(X, Y, Z, W ) = B(X, Y, Z, W )
3
where
B(X, Y, Z, W ) = H(X, Y, Z, W )
3
1
− {g(X, W )[ρe(Y, JZ) + ρe(JY, JZ)] + g(Y, Z)[ρe(X, JW ) + ρe(JX, W )]
8
−g(X, Z)[ρe(Y, JW ) + ρe(JY, W )] − g(Y, W )[ρe(X, JZ) + ρe(JX, Z)]
+F (X, W )[ρe(Y, Z) − ρe(JY, JZ)] + F (Y, Z)[ρe(X, W ) − ρe(JX, JW )]
−F (X, Z)[ρe(Y, W ) − ρe(JY, JW )] − F (Y, W )[ρe(X, Z) − ρe(JX, JZ)]}
+
e
κ
[g(X, W )F (Y, Z) + g(Y, Z)F (X, W )
4(n − 1)
−g(X, Z)F (Y, W ) − g(Y, W )F (X, Z)].
(6.1)
It is easy to see that the tensor (6.1) is the algebraic curvature tensor
and that it satisfies the condition of type (1.2).
The tensor (6.1) is the third conformally invariant algebraic curvature
tensor of an almost anti-Hermitian manifold. Its first Ricci tensor, in view
of (2.5), is
ρ( B)(Y, Z) =
3
∑
i
B(ei , Y, Z, ei )
3
1
= {ρ(Y, Z) − ρ(JY, JZ) − (n − 1)[ρe(Y, JZ) + ρe(JY, Z)]
4
−g(Y, Z)
∑
}
(6.2)
e F (Y, Z) ,
ρe(ei , Jei ) + κ
i
such that, according (3.14), we have
1
ρ( B)(Y, Z) = P ′′ (Y, Z).
4
3
(6.3)
54
Mileva Prvanović
As for the second Ricci, we have
ρe( B)(Y, Z) =
3
∑
i
B(Jei , Y, Z, ei )
3
1
= {ρ(Y, JZ) + ρ(JY, Z) + (n − 1)[ρe(Y, Z) − ρe(JY, JZ)]
4
e g(Y, Z) − F (Y, Z)
−κ
∑
}
(6.4)
ρe(ei , Jei ) ,
i
and therefore
1
ρe( B)(Y, Z) = ρ( B)(JY, Z) = P ′′ (Y, JZ).
4
3
3
(6.5)
Thus, we can state
Theorem 6.1 For an almost anti-Hermitian manifold, the tensor (6.1)
is the third conformally invariant algebraic curvature tensor. The relations
(6.2) and (6.4) determine its Ricci tensors and the relations (6.3) and (6.5)
hold.
If (M, g, J) is an anti-Kähler manifold, then
B(X, Y, Z, W ) = R(X, Y, Z, W )
3
1
+ {g(X, W )ρ(Y, Z) + g(Y, Z)ρ(X, W )
4
−g(X, Z)ρ(Y, W ) − g(Y, W )ρ(X, Z)
−F (X, W )ρ(Y, JZ) − F (Y, Z)ρ(X, JW )
+F (X, Z)ρ(Y, JW ) + F (Y, W )ρ(X, JZ)}
+
e
κ
[g(X, W )F (Y, Z) + g(Y, Z)F (X, W )
4(n − 1)
−g(X, Z)F (Y, W ) − g(Y, W )F (X, Z)]
and
ρ( B)(Y, Z) =
3
n
1
e F (Y, Z)].
ρ(Y, Z) + [κg(Y, Z) + κ
2
4
Conformal change of the metric on almost anti-Hermitian manifolds
7. Linear dependence of the conformally invariant tensors
In view of (5.1), (6.1), (2.5) and (2.6), we have
1
[(n − 1) B(X, Y, Z, W ) − B(X, Y, Z, W )] =
n−2
2
3
1
H(X, Y, Z, W ) −
[g(X, W )ρ(H)(Y, Z)
2(n − 2)
+g(Y, Z)ρ(H)(X, W )
−g(X, Z)ρ(H)(Y, W ) − g(Y, W )ρ(H)(X, Z)
−F (X, W )ρe(H)(Y, Z) − F (Y, Z)ρe(H)(X, W )
+F (X, Z)ρe(H)(Y, W ) + F (Y, W )ρe(H)(X, Z)]
+
−
κ
[g(X, W )g(Y, Z) − g(X, Z)g(Y, W )
4(n − 2)(2n − 1)
−F (X, W )F (Y, Z) + F (X, Z)F (Y, W )]
e
κ
[g(X, W )F (Y, Z) + g(Y, Z)F (X, W )
4(n − 1)(n − 2)
−g(X, Z)F (Y, W ) − g(Y, W )F (X, Y )].
This, taking into account (4.4), can be rewritten in the form
1
[(n − 1) B(X, Y, Z, W ) − B(X, Y, Z, W )] = B(X, Y, Z, W )
n−2
2
3
1
[
]
1
κ(H)
κ
+
−
+
[g(X, W )g(Y, Z) − g(X, Z)g(Y, W )
4(n − 1)
n − 2 2n − 1
−F (X, W )F (Y, Z) + F (X, Z)F (Y, W )[
1
e (H) − κ
e ][g(X, W )F (Y, Z)
[κ
4(n − 1)(n − 2)
+g(Y, Z)F (X, W ) − g(X, Z)F (Y, W ) − g(Y, W )F (X, Z)],
+
55
56
Mileva Prvanović
or, according (2.8), in the form
1
[(n − 1) B(X, Y, Z, W ) − B(X, Y, Z, W )]
n−2
1
3 ]
[ 2
∑
κ
1
−
ρe(ei , Jei )
+
8(n − 1)(n − 2) 2n − 1
i
[g(X, W )g(Y, Z) − g(X, Z)g(Y, W ) − F (X, W )F (Y, Z) + F (X, Z)F (Y, W )]
B(X, Y, Z, W ) =
[
]
∑
1
e−
+
κ
ρ(ei , Jei )
8(n − 1)(n − 2)
i
[g(X, W )F (Y, Z) + g(Y, Z)F (X, W ) − g(X, Z)F (Y, W ) − g(Y, W )F (X, Z)].
(7.1)
If we put
[
]
∑
κ
B(X, Y, Z, W ) =
−
ρe(ei , Jei ) [g(X, W )g(Y, Z) − g(X, Z)g(Y, W )
2n − 1
4
i
−F (X, W )F (Y, Z) + F (X, Z)F (Y, W )],
(7.2)
and
[
e−
B(X, Y, Z, W ) = κ
∑
5
]
ρe(ei , Jei ) [g(X, W )F (Y, Z) + g(Y, Z)F (X, W )
i
−g(X, Z)F (Y, W ) − g(Y, W )F (X, Z)],
(7.3)
we find that, in view of (3.11) and (3.12), we have
e−2f B̄(X, Y, Z, W ) = B(X, Y, Z, W ),
4
and
4
e−2f B̄(X, Y, Z, W ) = B(X, Y, Z, W ).
5
5
Thus, the relation (7.1) can be expressed as follows
B=
1
1
1
[(n − 1) B − B] +
( B + B).
n−2
8(n − 1)(n − 2) 4
2
3
5
(7.4)
Each of the tensors B, . . . , B is the algebraic curvature tensor, and each
1
5
satisfies the condition of type (1.2). Thus, we can state the theorem
Theorem 7.1 The conformally invariant tensors B, . . . , B are linearly
1
5
Conformal change of the metric on almost anti-Hermitian manifolds
57
dependent such that the relation (7.4) holds. Each of this tensors is algebraic
curvature tensor and each satisfies the condition of type (1.2).
If (M, g, J) is and anti-Kähler manifold, then, according (1.6)
∑
κ
2n
−
ρe(ei , Jei ) =
κ,
2n − 1
2n
−1
i
e−
κ
∑
ρ(ei , Jei ) = 0.
i
Thus,
2n
κ[g(X, W )g(Y, Z) − g(X, Z)g(Y, W )
2n − 1
−F (X, W )F (Y, Z) + F (X, Z)F (Y, W )]
B(X, Y, Z, W ) =
4
and (7.4) reduces to
B=
1
n
1
[(n − 1) B − B] +
B,
n−2
4(n − 1)(n − 2) 4
2
3
where, now, B, B and B are given by the relation (4.5), (5.10) and (6.6)
respectively.
1
2
3
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